Primary & Middle School Mathematics · Grades 1–9

24First Fractions

Half an apple, a quarter of an hour, three quarters of the class: fractions name the pieces of things. This chapter introduces the writing $\frac{a}{b}$ , reads fractions on pictures and number lines, and compares the simplest ones. (Fractions grow up in Chapter 39.)

24.1 Naming the pieces

Definition 24.1 (Fraction)

Cut a whole into equal parts and take some — that is a fraction:

\frac{3}{4} \quad \begin{array}{l} 3 \text{: how many parts are taken (the \emph{numerator});}\\ 4 \text{: how many equal parts the whole is cut into (the \emph{denominator}).} \end{array}

$\frac12$ is a half, $\frac13$ a third, $\frac14$ a quarter, $\frac{1}{10}$ a tenth.

Reading fractions on pictures. The parts must be equal: three unequal pieces do not make thirds! — Reading fractions on pictures. The parts must be *equal*: three unequal pieces do not make thirds!

Example 24.2 (Fractions of a collection)

$\frac13$ of $12$ marbles: share the $12$ marbles into $3$ equal groups of $4$ ; one group is $\frac13$ , so $\frac13$ of $12$ is $4$ — and $\frac23$ of $12$ is two groups, $8$ .

24.2 Fractions on the number line

Method 24.3 (Placing $\frac{a}{b}$ )

Cut each unit of the line into $b$ equal steps; from $0$ , walk $a$ steps. If $a$ is bigger than $b$ , the walk passes $1$ : fractions can be bigger than one whole!

The line cut in quarters: 2/4 lands on the same point as 1/2; 5/4 is one quarter past 1; 11/4 is 2 and 3/4. — The line cut in quarters: $\frac24$ lands on the same point as $\frac12$ ; $\frac54$ is one quarter past $1$ ; $\frac{11}{4}$ is $2$ and $\frac34$ .

Example 24.4 (Whole numbers hiding in fractions)

$\frac44 = 1$ (four quarters make the whole); $\frac82 = 4$ ; $\frac{12}{3} = 4$ . And $\frac74$ is $1 + \frac34$ : one whole and three quarters more.

24.3 Comparing and adding simple fractions

Proposition 24.5 (Same denominator)

With the same denominator, the pieces have the same size — so just count them:

\frac{2}{8} < \frac{5}{8}, \qquad \frac{3}{8} + \frac{4}{8} = \frac{7}{8}.

Comparing to one whole is also easy: $\frac{5}{8} < 1 < \frac{9}{8}$ (fewer, then more, than $8$ eighths).

Why, on a picture. Shading $3$ eighths and then $4$ more eighths of the same bar shades $7$ eighths in total. ∎

Example 24.6 (Different denominators, same point)

On the number line, $\frac12$ , $\frac24$ and $\frac{5}{10}$ all land on the same point: they are three names of the same number. Cutting each half into two makes quarters; into five, tenths. (The general rule is in Chapter 39.)

Example 24.7 (Quarters of an hour)

An hour has $60$ minutes, so a quarter of an hour is $60 \div 4 = 15$ minutes, and three quarters of an hour is $45$ minutes. “Half past two” is $\frac12$ hour after two o’clock.

24.4 Exercises

Exercise 24.1 ★

Write the fraction shown: a pizza cut in $6$ with $5$ slices left; a chocolate bar of $8$ squares with $3$ eaten (fraction eaten); a flag divided in $3$ equal vertical bands, $1$ colored.

Solution

Solution of Exercise 24.1.

Pizza: $\frac56$ left. Chocolate: $\frac38$ eaten. Flag: $\frac13$ colored.

Exercise 24.2 ★

Draw a bar cut into $5$ equal parts and shade $\frac35$ . Then shade $\frac{2}{5}$ of another equal bar. Which shading is bigger?

Solution

Solution of Exercise 24.2.

$\frac35$ shades three parts, $\frac25$ only two: $\frac35$ is the bigger shading.

Exercise 24.3 ★

Compute: $\frac12$ of $18$ ; $\frac14$ of $20$ ; $\frac13$ of $21$ ; $\frac34$ of $20$ (three groups of a quarter).

Solution

Solution of Exercise 24.3.

Half of $18$ : $9$ . Quarter of $20$ : $5$ . Third of $21$ : $7$ . $\frac34$ of $20$ : three quarters, $3 \times 5 = 15$ .

Exercise 24.4 ★

Place on a number line cut in thirds: $\frac13$ ; $\frac33$ ; $\frac53$ ; $2$ . Which fraction lands exactly on $2$ ?

Solution

Solution of Exercise 24.4.

$\frac13$ : one step; $\frac33$ : on $1$ ; $\frac53$ : two steps past $1$ . The fraction landing exactly on $2$ is $\frac63$ .

Exercise 24.5 ★

Copy and complete with $<$ , $>$ or $=$ :

\frac38 \;?\; \frac68, \qquad \frac55 \;?\; 1, \qquad \frac74 \;?\; 1, \qquad \frac12 \;?\; \frac24 .

Solution

Solution of Exercise 24.5.

$\frac38 < \frac68$ ; $\frac55 = 1$ ; $\frac74 > 1$ ; $\frac12 = \frac24$ .

Exercise 24.6 ★

Compute: $\frac26 + \frac36$ ; $\frac58 - \frac28$ ; $\frac14 + \frac24$ ; $1 - \frac13$ (how many thirds make $1$ ?).

Solution

Solution of Exercise 24.6.

$\frac26 + \frac36 = \frac56$ ; $\frac58 - \frac28 = \frac38$ ; $\frac14 + \frac24 = \frac34$ ; $1 = \frac33$ , so $1 - \frac13 = \frac23$ .

Exercise 24.7 ★

Write as a whole number plus a fraction smaller than $1$ : $\frac54$ ; $\frac73$ ; $\frac{9}{2}$ .

Solution

Solution of Exercise 24.7.

$\frac54 = 1 + \frac14$ ; $\frac73 = 2 + \frac13$ ; $\frac92 = 4 + \frac12$ .

Exercise 24.8 ★

How many minutes are: half an hour; a quarter of an hour; three quarters of an hour; $\frac13$ of an hour?

Solution

Solution of Exercise 24.8.

Half an hour: $30$ min. Quarter: $15$ min. Three quarters: $45$ min. Third: $20$ min.

Exercise 24.9 ★

In a class of $24$ students, $\frac14$ wear glasses. How many students wear glasses? How many do not?

Solution

Solution of Exercise 24.9.

$\frac14$ of $24$ is $6$ students with glasses; $24 - 6 = 18$ without.

Exercise 24.10 ★★

Nina ate $\frac38$ of a pizza and Sam ate $\frac48$ of the same pizza. What fraction did they eat together? What fraction is left? Who ate more?

Solution

Solution of Exercise 24.10.

Together: $\frac38 + \frac48 = \frac78$ . Left: $\frac18$ . Sam ate more ( $4$ eighths against $3$ ).

Exercise 24.11 ★★

True or false, with a picture or an example: “ $\frac12$ of a small pizza is less than $\frac14$ of a very big pizza is possible”. What must one always know before comparing two fractions of something?

Solution

Solution of Exercise 24.11.

Possible indeed: half of a small pizza can be less food than a quarter of a giant one. Fractions compare parts of the same whole; before comparing $\frac12$ and $\frac14$ “of something”, one must know that the two somethings are equal.